Question

The subtangent, ordinate and subnormal to the parabola $y^2= 4ax$ at a point (different from the Origin) are in

• A. G.P
• B. A.P.
• C. H.P
• D. none of these

Answer 1

Answer: (A)

G.P

Answer 2

$y^{2} = 4ax$ $\Rightarrow 2\,y \frac{dy}{dx} = 4a$ $\Rightarrow \frac{dy}{dx} = \frac{2a}{y}$ Subtangent $= \frac{y}{dy / dx}=\frac{y}{2\, a/y}$ $= \frac{y^{2}}{2a} = \frac{4ax}{2a}= 2x$ Ordinate $= y = \sqrt{4\,ax} = 2\sqrt{a} \sqrt{x}$ Subnormal $= y \frac{dy}{dx} = 2a$ Since (subtangent) (subnormal) $= \left(2x\right)\left(2a\right)$ $= 4ax =$ (ordinate)$^{2}$ $\Rightarrow$ subtangent, ordinate and subnormal are in $G.P$.